There is no "winning country" in the Olympics. The games celebrate human achievement not national achievement. That being said, the US has won the most medals overall in all of the Olympics, Canada may well have won the most Gold Medals in a Winter Olympics and the US may well have won the most medals of any type in any of the Winter Olympics. I don't know which country has won the most Golds and most medals overall in the Summer Olympics.
Norway with 13 gold medals.
The previous record for gold medals by the host country was 10. The previous best for gold medals at the winter games, host or not, was 13 by Norway and Russia.
The USA because they had the most medals and gold medals.
Canada won most gold medals (14) overall in the 2010 Winter Olympics in Vancouver.
The country with the most gold medals at the Rio 2016 Olympics was the United States with 46 total.
So far the US won gold medals in certain events during the 2012 Olympics, but as to which country will win the most medals overall, that cannot be determined as of now.
Germany had the most gold medals at the 2006 Winter Olympics with 11.
Germany won the most gold medals at the 1998 Winter Olympics with 12.
Norway
The country who is hosting the Olympics
China.